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The problem is asking to demonstrate two properties of definite integrals related to even and odd functions.

### Solution

#### Part (a)
**Statement**: Show that if \( f \) is an even function, then
\[
\int_{-a}^{a} f(x) \, dx = 2 \int_0^{a} f(x) \, dx.
\]

1. **Definition of an even function**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.

2. **Set up the integral**:
   \[
   \int_{-a}^{a} f(x) \, dx = \int_{-a}^{0} f(x) \, dx + \int_{0}^{a} f(x) \, dx.
   \]

3. **Use the property of even functions**: Since \( f(x) \) is even, \( f(x) = f(-x) \). In the interval \([-a, 0]\), let \( u = -x \) (then \( du = -dx \)). When \( x = -a \), \( u = a \), and when \( x = 0 \), \( u = 0 \). Thus,
   \[
   \int_{-a}^{0} f(x) \, dx = \int_{a}^{0} f(-u) \, (-du) = \int_{0}^{a} f(u) \, du = \int_{0}^{a} f(x) \, dx.
   \]

4. **Combine integrals**:
   \[
   \int_{-a}^{a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx.
   \]
Thus, the result is proven.

#### Part (b)
**Statement**: Show that if \( f \) is an odd function, then
\[
\int_{-a}^{a} f(x) \, dx = 0.
\]

1. **Definition of an odd function**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.

2. **Set up the integral**:
   \[
   \int_{-a}^{a} f(x) \, dx = \int_{-a}^{0} f(x) \, dx + \int_{0}^{a} f(x) \, dx.
   \]

3. **Use the property of odd functions**: Since \( f(x) \) is odd, \( f(-x) = -f(x) \). In the interval \([-a, 0]\), let \( u = -x \) (then \( du = -dx \)). When \( x = -a \), \( u = a \), and when \( x = 0 \), \( u = 0 \). Thus,
   \[
   \int_{-a}^{0} f(x) \, dx = \int_{a}^{0} f(-u) \, (-du) = -\int_{0}^{a} f(u) \, du = -\int_{0}^{a} f(x) \, dx.
   \]

4. **Combine integrals**:
   \[
   \int_{-a}^{a} f(x) \, dx = \int_{0}^{a} f(x) \, dx - \int_{0}^{a} f(x) \, dx = 0.
   \]
Thus, the result is proven.

Would you like more details on any of these steps, or do you have further questions?

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Here are 5 related questions to expand your understanding:
1. What are some examples of even functions and odd functions?
2. How would these properties change if \( f(x) \) were neither even nor odd?
3. Why is the integral of an odd function over a symmetric interval always zero?
4. Can you apply these properties to solve definite integrals more efficiently?
5. How do these properties apply in Fourier analysis for decomposing functions?

**Tip:** When calculating definite integrals over symmetric intervals, check if the function is even or odd. This can simplify the calculation significantly.

Montrer que si f est une fonction paire, alors ∫[-a, a] f(x) dx = 2 ∫[0, a] f(x) dx. Montrer que si f est une fonction impaire, alors ∫[-a, a] f(x) dx = 0.

Explore the properties of definite integrals for even and odd functions. Learn how to show that for even functions, the integral over a symmetric interval equals twice the integral over half the interval, and for odd functions, the integral over a symmetric interval is zero.

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